Combinatorial decompositions of a class of rings

Abstract

In this paper we introduce some combinatorial methods for the study of graded rings and use these methods to analyze a class of rings that are of importance in combinatorics. Our methods are based on a “unique representation theorem” which is shown (Section 2) to hold for any graded algebra and which in a sense is an extension of the direct sum decomposition for Cohen-Macaulay algebras. This unique representation is reminiscent of a similar result obtained by Rees [20]. Both results yield a decomposition of the algebra. However, our result differs from that of Rees in that it carries more information about the algebra. Our main results are concerned with a class of rings (see Section 3 for the definition) associated to simplicial complexes. There are several natural operations that can be performed on simplicial complexes to obtain new complexes which in turn have an interpretation for the corresponding rings. We show (Sections 4, 5, and 6) that our basic decompositions can be transferred during these operations from the original rings to the newly constructed ones. The operations studied here are: the chain transform or barycentric subdivision (Section 4), rank-selection both for partially ordered sets and for simplicial complexes (Section S), and localization (Section 6). As we develop these tools we give some applications. In Section 5 we give a new topological characterization of the Cohen-Macaulay property for partially ordered sets, an immediate consequence of which is the RankSelection Theorem (Baclawski [2]). This characterization is closely related to those obtained by Garsia in [14].